Lesson Worksheet: Binomial Theorem: Using Partial Fractions Mathematics

In this worksheet, we will practice decomposing rational expressions into partial fractions, then expanding them using the binomial theorem.

Q1:

By decomposing 3𝑥1(𝑥+1)(𝑥3) into partial fractions, find the quadratic approximation of 3𝑥1(𝑥+1)(𝑥3) and state the range of values of 𝑥 for which the approximation is valid.

  • A13119𝑥2527𝑥,|𝑥|<1.
  • B13119𝑥+2527𝑥, |𝑥|<3.
  • C13119𝑥+2527𝑥, |𝑥|<1.
  • D13+119𝑥+2527𝑥, |𝑥|<1.
  • E13+119𝑥+2527𝑥, |𝑥|<3.

Q2:

By decomposing 3𝑥(1𝑥)(𝑥+2) into partial fractions, find the cubic approximation of 3𝑥(1𝑥)(𝑥+2) and state the range of values of 𝑥 for which the approximation is valid.

  • A32𝑥+34𝑥+98𝑥, |𝑥|<2
  • B32𝑥34𝑥+98𝑥, |𝑥|<2
  • C32𝑥34𝑥98𝑥,|𝑥|<1
  • D32𝑥34𝑥+98𝑥, |𝑥|<1
  • E32𝑥+34𝑥+98𝑥, |𝑥|<1

Q3:

By expressing 𝑥3(𝑥+2) in partial fractions, find the cubic expansion of 𝑥3(𝑥+2) and state the range of values of 𝑥 for which the approximation is valid.

  • A34+12𝑥516𝑥+316𝑥, |𝑥|<2
  • B34+12𝑥516𝑥+316𝑥, |𝑥|<1
  • C34+12𝑥516𝑥+316𝑥, |𝑥|<1
  • D34+12𝑥516𝑥+316𝑥, |𝑥|<2
  • E34+12𝑥516𝑥+316𝑥, 𝑥<2

Q4:

By expressing 1+2𝑥+5𝑥(𝑥+1)(𝑥+2)(𝑥1) in partial fractions, find the quadratic approximation of 1+2𝑥+5𝑥(𝑥+1)(𝑥+2)(𝑥1) and state the range of values of 𝑥 for which the approximation is valid.

  • A12+114𝑥+78𝑥, |𝑥|<1
  • B12114𝑥+78𝑥, |𝑥|<2
  • C12114𝑥+78𝑥, |𝑥|<1
  • D12+114𝑥+78𝑥, |𝑥|<2
  • E12+114𝑥78𝑥, |𝑥|<1

Q5:

By writing 𝑥+3(2𝑥+1)(13𝑥) in the form 𝐴2𝑥+1+𝐵13𝑥, find the expansion of 𝑥+3(2𝑥+1)(13𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A3+4𝑥+22𝑥, |𝑥|<1
  • B34𝑥+22𝑥, |𝑥|<12
  • C34𝑥+22𝑥, |𝑥|<13
  • D3+4𝑥+22𝑥, |𝑥|<13
  • E3+4𝑥+22𝑥, |𝑥|<12

Q6:

By decomposing 𝑥𝑥4(𝑥2)(𝑥3) into partial fractions, find the quadratic approximation of 𝑥𝑥4(𝑥2)(𝑥3) and state the range of values of 𝑥 for which the approximation is valid.

  • A231318𝑥35108𝑥, |𝑥|<2
  • B531318𝑥35108𝑥, |𝑥|<3
  • C531318𝑥35108𝑥, |𝑥|<2
  • D11318𝑥35108𝑥, |𝑥|<2
  • E231318𝑥35108𝑥, |𝑥|<3

Q7:

By writing 4𝑥+15𝑥5(𝑥+3)(2𝑥1) in the form 𝐴+𝐵𝑥+3+𝐶2𝑥1, find the expansion of 4𝑥+15𝑥5(𝑥+3)(2𝑥1) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A53209𝑥10627𝑥, |𝑥|<3
  • B53209𝑥+10627𝑥, |𝑥|<12
  • C53209𝑥10627𝑥, |𝑥|<12
  • D13209𝑥10627𝑥, |𝑥|<3
  • E13209𝑥10627𝑥, |𝑥|<12

Q8:

By writing 𝑓(𝑥)=10𝑥6𝑥+2(𝑥+1)(2𝑥1) in the form 𝐴2𝑥1+𝐵(2𝑥1)+𝐶𝑥+1, find the expansion of 10𝑥6𝑥+2(𝑥+1)(2𝑥1) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A2+4𝑥+10𝑥+22𝑥, |𝑥|<12
  • B2+10𝑥+22𝑥, |𝑥|<1
  • C2+4𝑥+10𝑥+22𝑥, |𝑥|<1
  • D2+10𝑥+22𝑥, |𝑥|<12
  • E1+10𝑥+22𝑥, |𝑥|<12

Find the percentage error in using approximation to estimate the value of 𝑓(0.1), giving your answer to 2 decimal places.

Q9:

By decomposing 𝑓(𝑥)=𝑥𝑥1(𝑥1) into partial fractions, find the expansion of 𝑓(𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A1+𝑥+2𝑥+3𝑥, |𝑥|<1
  • B3𝑥4𝑥5𝑥, |𝑥|<1
  • C1𝑥2𝑥3𝑥, |𝑥|<1
  • D13𝑥4𝑥5𝑥, |𝑥|<1
  • E1+3𝑥+4𝑥+5𝑥, |𝑥|<1

Find the percentage error in using approximation to estimate the value of 𝑓12.

Q10:

By decomposing 𝑓(𝑥)=𝑥+𝑥+1(𝑥+1) into partial fractions, find the expansion of 𝑓(𝑥) up to the 𝑥 term and state the values of 𝑥 for which the expansion is valid.

  • A14𝑥+8𝑥13𝑥, |𝑥|<1
  • B14𝑥+4𝑥7𝑥, |𝑥|<1
  • C12𝑥+4𝑥7𝑥, |𝑥|<1
  • D1+2𝑥4𝑥+7𝑥, |𝑥|<1
  • E1+4𝑥8𝑥+13𝑥, |𝑥|<1

Find the percentage error in using the approximation to estimate the value of 𝑓13, giving your answer to one decimal place.

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